{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# TauVector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.741577Z",
     "start_time": "2021-02-15T09:50:56.083243Z"
    }
   },
   "outputs": [],
   "source": [
    "from fractions import Fraction\n",
    "import poisson_approval as pa"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Tau-Vector and its Basic Properties"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The most usual way to define a tau-vector (ballot shares) is to deduce it from applying a given strategy in a given profile, like in the tutorial on *ProfileNoisyDiscrete*. That being said, a tau-vector can also be defined directly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.752547Z",
     "start_time": "2021-02-15T09:51:01.745566Z"
    }
   },
   "outputs": [],
   "source": [
    "tau = pa.TauVector({'a': Fraction(1, 10), 'ab': Fraction(3, 5), 'c': Fraction(3, 10)})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Share of the ballot $ab$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.788977Z",
     "start_time": "2021-02-15T09:51:01.758530Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fraction(3, 5)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.ab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Share of single / double votes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.810947Z",
     "start_time": "2021-02-15T09:51:01.791969Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fraction(2, 5)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.share_single_votes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.828899Z",
     "start_time": "2021-02-15T09:51:01.812941Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fraction(3, 5)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.share_double_votes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scores:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.851837Z",
     "start_time": "2021-02-15T09:51:01.830894Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{a: 7/10, b: 3/5, c: 3/10}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.scores"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Winners:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.875773Z",
     "start_time": "2021-02-15T09:51:01.854842Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'a'}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.winners"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Events: Duos, Pivots and Trios"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An *Event* provides the asymptotic development of the probability when the expected number of voters $n$ tend to infinity, as well as the offsets for each type of ballot."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Duo between $a$ and $b$:\n",
    "\n",
    "($a$ and $b$ have the same score, but not necessarily higher than $c$.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.893725Z",
     "start_time": "2021-02-15T09:51:01.879764Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.duo_ab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Access the asymptotic development and the offsets:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.913673Z",
     "start_time": "2021-02-15T09:51:01.896717Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(- 0.1 n + o(1))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.duo_ab.asymptotic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.930626Z",
     "start_time": "2021-02-15T09:51:01.915666Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.duo_ab.phi_a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pivots between $a$ and $b$:\n",
    "\n",
    "(For the definition of each variant, cf. the *Reference* section.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.950575Z",
     "start_time": "2021-02-15T09:51:01.932621Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_weak_ab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:01.987477Z",
     "start_time": "2021-02-15T09:51:01.953566Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_strict_ab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.017402Z",
     "start_time": "2021-02-15T09:51:01.991464Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_tij_abc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.048653Z",
     "start_time": "2021-02-15T09:51:02.021384Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- inf)>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_tjk_abc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Trios:\n",
    "\n",
    "(For the definition of each variant, cf. the *Reference* section.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.083608Z",
     "start_time": "2021-02-15T09:51:02.051651Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.trio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.115473Z",
     "start_time": "2021-02-15T09:51:02.087548Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- inf)>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.trio_1t_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.145657Z",
     "start_time": "2021-02-15T09:51:02.118465Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- 0.151472 n - 0.5 log n - 1.18339 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.trio_2t_ab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Print the order of the magnitudes of the pivots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.187016Z",
     "start_time": "2021-02-15T09:51:02.149645Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mu_ac > mu_ab > mu_bc\n"
     ]
    }
   ],
   "source": [
    "tau.print_magnitudes_order()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Print all the weak pivots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.218461Z",
     "start_time": "2021-02-15T09:51:02.189539Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pivot_weak_ab:  <asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_weak_ac:  <asymptotic = exp(- 0.0834849 n - 0.5 log n - 0.87535 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_weak_bc:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "trio:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n"
     ]
    }
   ],
   "source": [
    "tau.print_weak_pivots()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Print all the pivots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.264528Z",
     "start_time": "2021-02-15T09:51:02.222451Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pivot_weak_ab:  <asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_weak_ac:  <asymptotic = exp(- 0.0834849 n - 0.5 log n - 0.87535 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_weak_bc:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "pivot_strict_ab:  <asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_strict_ac:  <asymptotic = exp(- 0.0834849 n - 0.5 log n - 0.87535 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_strict_bc:  <asymptotic = exp(- inf)>\n",
      "pivot_tij_abc:  <asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_tij_acb:  <asymptotic = exp(- 0.0834849 n - 0.5 log n - 0.371758 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_tij_bac:  <asymptotic = exp(- 0.1 n + log n - 2.30259 + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_tij_bca:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.302013 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "pivot_tij_cab:  <asymptotic = exp(- 0.0834849 n - 0.5 log n + 0.0518905 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_tij_cba:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "pivot_tjk_abc:  <asymptotic = exp(- inf)>\n",
      "pivot_tjk_acb:  <asymptotic = exp(- inf)>\n",
      "pivot_tjk_bac:  <asymptotic = exp(- 0.0834849 n - 0.5 log n - 0.371758 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_tjk_bca:  <asymptotic = exp(- 0.0834849 n - 0.5 log n + 0.0518905 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_tjk_cab:  <asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_tjk_cba:  <asymptotic = exp(- 0.1 n + log n - 2.30259 + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "trio:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "trio_1t_a:  <asymptotic = exp(- inf)>\n",
      "trio_1t_b:  <asymptotic = exp(- 0.151472 n + 0.5 log n - 3.48597 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "trio_1t_c:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.490239 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "trio_2t_ab:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 1.18339 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "trio_2t_ac:  <asymptotic = exp(- inf)>\n",
      "trio_2t_bc:  <asymptotic = exp(- 0.151472 n + 0.5 log n - 3.1394 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "duo_ab:  <asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "duo_ac:  <asymptotic = exp(- 0.0834849 n - 0.5 log n - 0.87535 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "duo_bc:  <asymptotic = exp(- 0.0514719 n - 0.5 log n - 0.836813 + o(1)), phi_a = 1, phi_c = 1.41421, phi_ab = 0.707107>\n"
     ]
    }
   ],
   "source": [
    "tau.print_all_pivots()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Best Responses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The *focus* of a tau-vector is defined as:\n",
    "\n",
    "* *Direct*: the three weak pivots have distinct magnitudes.\n",
    "* *Forward-Focused*: the two lowest magnitudes are equal, the highest one is different.\n",
    "* *Backward-Focused*: the two highest magnitudes are equal, the lowest one is different.\n",
    "* *Unfocused*: the three magnitudes are equal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.283478Z",
     "start_time": "2021-02-15T09:51:02.269515Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Focus.DIRECT"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.focus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Best response for each ranking:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.326446Z",
     "start_time": "2021-02-15T09:51:02.287529Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{abc: <ballot = a, utility_threshold = 1, justification = Asymptotic method>, acb: <ballot = a, utility_threshold = 1, justification = Asymptotic method>, bac: <ballot = ab, utility_threshold = 0, justification = Asymptotic method>, bca: <ballot = bc, utility_threshold = 0, justification = Asymptotic method>, cab: <ballot = c, utility_threshold = 1, justification = Asymptotic method>, cba: <ballot = bc, utility_threshold = 0, justification = Asymptotic method>}"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.d_ranking_best_response"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Characteristics of the best response for voters with ranking $abc$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.356286Z",
     "start_time": "2021-02-15T09:51:02.333344Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<ballot = a, utility_threshold = 1, justification = Asymptotic method>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.d_ranking_best_response['abc']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.388238Z",
     "start_time": "2021-02-15T09:51:02.361270Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'a'"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.d_ranking_best_response['abc'].ballot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.427625Z",
     "start_time": "2021-02-15T09:51:02.392186Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.d_ranking_best_response['abc'].utility_threshold"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.466486Z",
     "start_time": "2021-02-15T09:51:02.431579Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Asymptotic method'"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.d_ranking_best_response['abc'].justification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.510369Z",
     "start_time": "2021-02-15T09:51:02.470475Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.d_ranking_best_response['abc'].pivot_tij"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Strategy with Attached Profile and Implicit Tau"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As mentioned in the tutorial on *ProfileNoisyDiscrete*, you can define a strategy with an attached profile:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.550749Z",
     "start_time": "2021-02-15T09:51:02.514358Z"
    }
   },
   "outputs": [],
   "source": [
    "profile = pa.ProfileNoisyDiscrete({\n",
    "    ('abc', 0.4, 0.01): Fraction(1, 10),\n",
    "    ('bac', 0.2, 0.01): Fraction(6, 10),\n",
    "    ('cab', 0.7, 0.01): Fraction(3, 10)\n",
    "})\n",
    "strategy = pa.StrategyOrdinal({'abc': 'a', 'bac': 'ab', 'cab': 'c'}, profile=profile)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This defines a tau vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.568705Z",
     "start_time": "2021-02-15T09:51:02.554740Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<a: 1/10, ab: 3/5, c: 3/10> ==> a"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "strategy.tau"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In that case, the strategy has shortcuts for all the properties of the tau vector. Thus for example, instead of writing:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.579671Z",
     "start_time": "2021-02-15T09:51:02.571909Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{a: 7/10, b: 3/5, c: 3/10}"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "strategy.tau.scores"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can simply write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.590817Z",
     "start_time": "2021-02-15T09:51:02.581666Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{a: 7/10, b: 3/5, c: 3/10}"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "strategy.scores"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, instead of writing:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.605602Z",
     "start_time": "2021-02-15T09:51:02.593634Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pivot_weak_ab:  <asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_weak_ac:  <asymptotic = exp(- 0.0834849 n - 0.5 log n - 0.87535 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_weak_bc:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "trio:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n"
     ]
    }
   ],
   "source": [
    "strategy.tau.print_weak_pivots()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.616573Z",
     "start_time": "2021-02-15T09:51:02.609592Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pivot_weak_ab:  <asymptotic = exp(- 0.1 n + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_weak_ac:  <asymptotic = exp(- 0.0834849 n - 0.5 log n - 0.87535 + o(1)), phi_a = 0.654654, phi_c = 1.52753, phi_ab = 0.654654>\n",
      "pivot_weak_bc:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n",
      "trio:  <asymptotic = exp(- 0.151472 n - 0.5 log n - 0.836813 + o(1)), phi_a = 0, phi_c = 1.41421, phi_ab = 0.707107>\n"
     ]
    }
   ],
   "source": [
    "strategy.print_weak_pivots()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbolic Computation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Up to now, all the computations we made were numeric. When defining a profile or a tau-vector, you can use the option ``symbolic=True``. In that case, all computations will be symbolic if possible. Note, however, that this option is much slower than numeric computation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:02.630207Z",
     "start_time": "2021-02-15T09:51:02.620211Z"
    }
   },
   "outputs": [],
   "source": [
    "profile = pa.ProfileNoisyDiscrete({\n",
    "    ('abc', 0.4, 0.01): Fraction(1, 10),\n",
    "    ('bac', 0.2, 0.01): Fraction(6, 10),\n",
    "    ('cab', 0.7, 0.01): Fraction(3, 10)\n",
    "}, symbolic=True)\n",
    "strategy = pa.StrategyOrdinal({'abc': 'a', 'bac': 'ab', 'cab': 'c'}, profile=profile)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-15T09:51:04.500063Z",
     "start_time": "2021-02-15T09:51:02.633199Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pivot_weak_ab:  <asymptotic = exp(- n/10 + o(1)), phi_a = 0, phi_c = 1, phi_ab = 1>\n",
      "pivot_weak_ac:  <asymptotic = exp(n*(-1 + sqrt(21)/5) - log(n)/2 - log(2*sqrt(21)*pi/5)/2 + o(1)), phi_a = sqrt(21)/7, phi_c = sqrt(21)/3, phi_ab = sqrt(21)/7>\n",
      "pivot_weak_bc:  <asymptotic = exp(n*(-1 + 3*sqrt(2)/5) - log(n)/2 - log(6*sqrt(2)*pi/5)/2 + o(1)), phi_a = 0, phi_c = sqrt(2), phi_ab = sqrt(2)/2>\n",
      "trio:  <asymptotic = exp(n*(-1 + 3*sqrt(2)/5) - log(n)/2 - log(6*sqrt(2)*pi/5)/2 + o(1)), phi_a = 0, phi_c = sqrt(2), phi_ab = sqrt(2)/2>\n"
     ]
    }
   ],
   "source": [
    "strategy.print_weak_pivots()"
   ]
  }
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